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Journal of Applied Mechanics, Vol. 10, No. 2, pp. 615-623, 2017. Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645. DOI: 10.18869/acadpub.jafm.73.239.26612

Ferrofluid Appendages: Fluid Fins, a Numerical Investigation on the Feasibility of using as Shapeable Propulsive Appendages

M. A. Feizi Chekab and P. Ghadimi†

Department of Marine Technology, Amirkabir University of Technology, Tehran, Tehran 14717, Iran

†Corresponding Author Email: [email protected]

(Received June 1, 2016; accepted November 19, 2016)

ABSTRACT

The present study focuses on the feasibility of using fluids, and in particular magnetic fluids, as “Fluid Structures” in designing external appendages for the submerged bodies and propulsive fins as a practical example. After reviewing the literature of the mathematical simulation of magnetic fluids and their applications, the concept of “Fluid Structures” and “Fluid Fins” are briefly introduced. The validation of the numerical solver against analytical solutions is presented and acceptable error of 1.21% up to 2.29% is estimated. Subsequently, the initial shaping of the as an external fluid fin, using three combinations of internal magnetic actuators, is presented which makes the way to the oscillating motion of the obtained fin, by producing a periodically changing . It is demonstrated that the shape of the fluid fin is almost the replica of the magnetic field. On the other hand, it is illustrated that a fluid fin with a size under 0.005 m on a circular submerged body of 1cm diameter could produce 0.0158 N force which is a high thrust force relative to the size of the body and the fin. Based on the obtained results, one may conclude that, when a “Fluid Fin” is capable of producing this amount of thrust, other fluid appendages could be scientifically contemplated and practically designed. Keywords: Fluid structures; Fluid appendages; Fluid fin; Ferrofluid; Numerical analysis.

NOMENCLATURE

B magnetic induction R rod radius D electric displacement t time E τ viscous Femag electromagnetic forces χ fluid magnetic sensitivity H magnetic field J ρ fluid density M fluid ρe electric p Δh elevation of the ferrofluid U fluid velocity μ0 magnetic permeability

1. INTRODUCTION in the field of in hydrodynamics is to make adaptive devices which could change shapes in response to Any device has a purpose for which a certain different circumstances of the surrounding fluid, an physical requirements are required. One of these expectation which is hard to achieve with requirements is the geometry of the device and its materials. The main idea of the present study is to structural strength to maintain the geometry, numerically assess the feasibility of replacing some preventing the device from cracking or breaking. “solid structures” with “fluid structures”. To make This is why solid materials are generally used to the idea more practical, in the present study, the fabricate such devices. However, the original reason practicability of this idea in hydrodynamics is behind breaking is nothing but the fact that the numerically demonstrated by the analysis of a device is originally solid and solid materials break. “Fluid Fin”, made of ferrofluid, as a propulsive Meanwhile, one of the main concerns of engineers appendage. M. A. Feizi Chekab and P. Ghadimi / JAFM, Vol. 10, No. 2, pp. 615-623, 2017.

Nowadays, are used in many fields including micro-pumps and micro-fluidic devices (Hartshorne et al., 2004 and Link et al. 2006), electro-magnetic control of droplet (Long et al., 2009; Nguyen et al., 2009 and Afkhami et al. 2008), (Sheikholeslami et al., 2015; Khan et al., 2015; Laird et al., 2004), nanotechnologies (Yellen et al., 2004 and Zahn, 2001), (Lajevardi et al., 2010, Malekzadeh et al., 2015 and Prakash, 2013), and many other industrial fields (Rosenweig, 2013). Electronic arts sector is another field of application of ferrofluids where ferrofluid sculptures (Kodama, 2008), ferrofluid displays and interfaces (Koelman et al., 2015 and Frey, 2004) have been produced (shown in Fig. 1). Fig. 2. Schematic of a fluid fin.

As observed in Fig. 2, magnetic actuators are located inside a submerged body and control the ferrofluid located outside the body. The ferrofluid is shaped in a way to form a fin-shaped structure. Subsequently, an oscillatory motion is induced to the fluid fin to make propulsion force.

(a) (b) By showing the feasibility of a fluid fin as a propulsion system, the authors have tried to show the possibility of using fluids in a wider range of applications. As a consequence of this study, one may imagine the replacement of solid materials with fluids in devices where rigid and solid parts limit the performance of the device. Fluid structures may include “fluid rudders”, “fluid walls and reservoirs”, “fluid propellers” and even “fluid skin” for “shape changing devices”. Also, by using electrically conducting ferrofluids, it is even possible to design fluid as shapeable and

(c) (d) changeable electrical circuits, perhaps resulting in Fig. 1. a) Fluid sculptures (Kodama, 2008), b & “flutronics” as a new concept in mechatronics. c) ferrofluid display (Koelman et al., 2015 and In the next section, the governing equations which Frey, 2004), d) ferrofluid interface are numerically solved by Ansys-CFX software are (Koh et al., 2011). presented. Afterwards, the software’s ability to simulate the ferrohydrodynamic is assessed by The scientific basis of Ferrohydrodynamics was modeling the stable state of a ferrofluid around a experimentally investigated by Shliomis et al. vertical . Then, the main (1980) and Cowley et al. (1967) who focused on the investigation is presented in the two above free surface stability of ferrofluids. Also, the mentioned parts. equations governing ferrofluids motion named ferrohydrodynamics equations, were initially presented by Resler and Rosensweig (1964) and 2. GOVERNING EQUATION Shliomis (1968). In this numerical study, the indtended simulations Different aspects of ferrofluid flows were analytically are conducted using the Ansys-CFX solver. The and numerically investigated by Shliomis et al. governing equations used in the numerical solver (1988) and Shliomis (2001a, b) who focused on are briefly presented in this section (Ansys, 2015). improving the equations of ferrohydrodynamics and The continuity of mass and are as ferrofluids magnetization and relaxation behavior, follows: Lavrora (2006) who presented numerical methods for ∂ρ predicting static axisymmetric shapes of ferrofluids +∇ρU = 0. (1) and Rabbi et al. (2016) who numerically studied the ∂t heat transfer of ferrofluids in a lid-driven cavity, ∂ρU among many others. +∇ ρU U = ∇p +∇ τ + F . (2) ∂t emag The aim of the present study is to numerically demonstrate the feasibility of using a , where U is the fluid velocity vector and t, p, and ρ specifically a ferrofluid as a small propulsive are the time, pressure, viscous stress, and fluid appendage resembling a fish tail or fin. The main density, respectively. Also, Femag represents the idea is better illustrated in Fig. 2. momentum due to electromagnetic forces. The

616 M. A. Feizi Chekab and P. Ghadimi / JAFM, Vol. 10, No. 2, pp. 615-623, 2017. complexity of the problem is attributed to the use of Maxwell equations as in ∂B ∇×E= , (3) ∂t ∂D ∇×H = J + , (4) ∂t

∇.B=0, (5)

∇ D= ρe , (6)

to calculate Femag. In Eq. (3) through (6), Fig. 4. Response of a magnetic fluid to the steady electric current passing through a vertical rod parameters B, D, E, H, J and ρe are magnetic induction, electric displacement, electric field, (Rosenweig, 1982).

magnetic field, current density, and density, respectively. Using these equations, the Figure 4 illustrates an experimental result of the equation problem. The analytical solution of the stable F = μ M∇H. (7) elevation of the ferrofluid is presented by emag 0 (Rosenweig, 1982). is extracted for Femag. In Eq. (7), M is the fluid μ χI 2 magnetization, H is the magnetic field and is the h = 0 (8) magnetic permeability of the vacuum. The above 8π 2 ρgR2 momentum source assumes that free surface is stable. In other words, the surrounding fluid, the In Eq. (8), Δh is the elevation of the ferrofluid, μ0 ferrofluid, and the magnetic field are set in a way that no instability occurs on the free surface. The is the magnetic permeability of the vaccum, χ is differences between stable and instable free the fluid magnetic sensitivity, I is the electric surfaces are displayed in Fig. 3. current, while ρ , g and R are the ferrofluid density, gravitational acceleration, and the rod radius, respectively. The setup of the problem is displayed in Fig. 5.

Fig. 3. Stable and instable free surface in ferrofluids.

The focus of the present study is on the stable free surface, where no spikes occur. The Finite Volume method is used by Ansys-CFX to solve the presented governing equations for the ferrohydrodynamics problems. After this brief presentation of the governing equations, the validation of the ferrohydrodynamics software is Fig. 5. Modeling of the validation problem. presented in the following section.

3. VALIDATION In the present simulation, the current is assumed to be 3.5 A, the rod radius is set to be 1mm, the 3 In order to validate the numerical solver, the ferrofluid density is 997 kg/m , and the magnetic ascension of ferrofluids along a vertical electrical sensitivity is varied as presented in Table 1 along current carrying rod is modeled and results are with the obtained results for each case. compared against analytical solutions (Rosenweig, As observed in Table 1, the elevation obtained for 1982). the three cases is very close to the analytical To better describe the problem, a magnetic field is solution and the error of the simulation ranges created around any rod carrying electrical current. from 1.21% to 2.29% which is acceptable for most The magnetic field attracts the ferrofluid which engineering simulations. The ferrofluid free ascends along the rod and reaches an equilibrium surface after solution is presented in Fig. 6 for test state due to gravitational force. case 1.

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Table 1 Input values and obtained results for the Also, three combinations of the magnetic actuators validation of the software in different cases are assumed as shown in Fig. 8. Dh numerical (mm) Dh analytical (mm) Test # error X

(a) 1 50 0.9869 1.0059 1.92% 2 75 1.4804 1.4625 1.21% 3 100 1.9739 2.019 2.29%

(b)

Fig. 6. Free surface elevation obtained for test case 1.

4. RESULTS AND DISCUSSION

As pointed out earlier, two magnetic actuators are placed in a submerged body to control the ferrofluid (c) shape and oscillatory motion in the current study. To this end, a computational domain with appropriate boundary conditions is required. Figure 3 shows the domain and numerical setup of the targeted analyses. Also, the boundary conditions are presented in Table 2. Fig. 8. Combination of magnetic actuators.

As observed in Fig. 8, the displayed combinations are essentially different to provide a better overview of the fluid fins produced in different combinations. In other words, in the present study, a preliminary study is conducted on the feasibility of the concept and not on the optimization of the system. Combination (a) is a pseudo 2D analysis, where the fluid fin sides are not free. In other words, the magnetic actuators are extended to the side walls, preventing the water to flow freely around the fin. Combination (b) is involved with four actuators to show the flexibility of the fluid fin shaping with the magnetic field. As observed in both combinations of (a) and (b), the magnetic actuators are extended

Fig. 7. Numerical domain and boundary to the walls. However, in combination (c), the sides conditions. of the ferrofluid are free, i.e. both actuators are in the middle of the body and at a distance from the walls, allowing the water to flow around the Table 2 Setup boundary conditions ferrofluid. Momentum BCs Magnetic BCs After choosing the combinations of the magnetic Body No slip wall Normal field actuators, the first step to make a fluid fin is to Actuators - Normal field oblige the fluid to maintain a stable initial shape. In the next part, using the above mentioned numerical Far field Opening Zero field setups, the stable shape of the fluid fin in the three Symmetries Symmetry Normal Field combinations is discussed.

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4.1 Shaping Stage: the Initial Shape of the Combination (b) Fluid Fin To obtain the initial stable shape of the fluid fin, the magnetic actuators are activated and during a transient analysis, the actuators are left to shape the ferrofluid. In this process, which is the shaping stage of the fluid fin, the actuators are generating a steady magnetic field of 0.125 Tesla. The development process of the stable shaping of the ferrofluid in combination (a) is displayed in Fig. 9.

Combination (c)

Initial condition

Fig. 10. Initial stable shapes obtained for combinations (b) and (c).

It should be pointed out that in combination (b), the outer magnetic actuators applies one tenth of the magnetic field of the middle ones, allowing the t = 0.003 s form shown in Fig. 10. As observed above, after the shaping stage, different shapes of fins can be obtained. This is due to the fact that the ferrofluid free surface is shaped by the magnetic field. By comparing the initial stable shapes through the magnetic field isosurfaces for different combinations (Fig. 11), it is observed

that the fluid fin tends to duplicate the magnetic t = 0.005 s field isosurfaces.

Fluid Fin Magnetic Field

Combination (a)

t = 0.01 s

Combination (b)

(Stable shape)

t = 0.03 s

Combination (c)

Fig. 9. Shaping stage of the combination (a).

As observed in Fig. 9, the fluid quickly reaches its stable shape, a process which lasts under 0.07s for all considered combinations. Fig. 11. Comparison of isosurfaces of the Also, the initial stable shapes obtained for magnetic field and related fluid fin in different combinations (b) and (c) are shown in Fig 10. combinations.

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Therefore, through designing the magnetic field periods of magnetic field ) in Fig. 14. isosurface shapes, the form of the fluid fin can be accurately predicted and designed. So far, it is shown that the external shape of ferrofluids can be designed through altering the magnetic field. In addition, the very fast response of the fluid to the magnetic field (under 0.07s) shows the ability of using ferrofluids in devices where fast shape changing is required. This is a promising result for the “fluid structures” concept. As observed, the ferrofluid is now placed on the submerged body as an external appendage which could play the role of a control surface as well as a lift surface. By inducing specific motions to the fluid fin, hydrodynamic forces can be produced. 4.2 Actuation Stage: Motion of the Fluid Fin After finding the stable shape of the fluid fin, an oscillatory magnetic field with a frequency of 40 Hz is applied to the fluid by the magnetic actuators. Because of the similarity of the actuation stage for all combinations, this process is only described for the combination (a). The magnetic field of the two actuators in combination (a) versus time is presented in Fig. 12.

0.3 ) 0.25 T ( ld 0.2 ie Fig. 14. Water pressure with motion of the F ic 0.15 magnetic fluid (combination (a)) and the et n Actuator 1 g 0.1 a Actuator 2 motion direction of the fluid fin. M0.05 0 0 0.05 0.1 0.15 0.2 0.25 As observed in Fig. 14, when the fin ascends, the time (s) water pressure increases above and decreases Fig. 12. Magnetic field of the actuators Vs time. bellow the body, up to 15 Pa. The inverse occurs, when the fluid fin descends. It is expected that the By applying the mentioned magnetic field to the fluid velocity also changes with the pressure. The ferrofluid, the overlapping magnetic field of the velocity field is presented in Fig. 15 for the extreme actuators results in a magnetic field which is conditions of the fin oscillation. transferred from an actuator to another. This As expected, the fluid fin acts on the water as a changing magnetic field causes the ferrofluid to be solid fin and velocities up to 0.25 m/s are induced transferred from one actuator to the other and to the water from the fin. It should be pointed out oscillate. The motion of the fluid fin is illustrated in that in some areas near the fin, the velocity Fig. 13. increases up to 0.9 m/s which is a very high velocity produced by a 5 mm fin. Same results with slight differences can be found for other combinations. As observed in Fig. 15, the actuation stage makes the fluid fin act like a pseudo solid fin and to affect the velocity and pressure field around the body.

Fig. 13. Magnetic fluid motion with the magnetic The obtained results show that not only the shape of field actuation (combination (a)) and the motion the ferrofluid is fully controllable, but also the fluid direction of the fluid fin. shows enough rigidity to affect the surrounding fluid and induce pressure and velocity changes, all controllable via the magnetic actuators. This result As observed in Fig. 13, the actuation of the extends the use of the “fluid structures” concept to a magnetic field results in the oscillation of the wider range of applications. ferrofluids in a course which eventually can be defined by the placement, power, and frequency of However, the most important variable for a the actuators. propulsive appendage is its produced thrust which is presented in the next section. The pressure field change is illustrated (after 40

620 M. A. Feizi Chekab and P. Ghadimi / JAFM, Vol. 10, No. 2, pp. 615-623, 2017.

Fig. 17. Thrust in the shaping stage (reaching a stable fluid fin shape).

Fig. 15. Fluid velocity with motion of the Fig. 18. Lateral force produced by the fluid fin magnetic fluid (combination (a)) and the motion versus time. direction of the fluid fin. As expected and observed in Fig. 18, the lateral force oscillates around 0 N. a relatively similar 4.3 Fluid Fin Thrust behavior happens in other combinations. After The propulsion forces acting on the fluid fin in almost 30 periods, the forces become periodically different combinations are illustrated in Fig. 16. steady for all cases. Figure 19 illustrates the forces on the fluid fin.

Fig. 19. Thrust produced by the fluid fin in Fig. 16. Thrust produced by the fluid fin versus propulsion stage for different combinations. time in different combinations.

As expected and observed in Fig. 16, the shaping As shown in Fig. 18, the fluid fin produces an stage of different combinations have different average thrust of 0.0158 N in combination (a) durations. Also, the magnitude of the thrust which is a large value relative to the size of the produced by the three combinations is different. body. Also, for combinations (b) and (c), an average The combination (a) produces the highest thrust, thrust of 0.0028 N and 0.00733 N are respectively followed by the combination (c) and (b) in obtained. descending order. As observed in Fig. 16, it is also The main reason behind the low thrust of shown that, in the first part of the simulation related combination (c) is the fact that the water is free to to the shaping stage, no significant force is flow around the ferrofluid; hence, a pressure release generated. occurs and the thrust diminishes. However, as observed in Fig. 17, as soon as the However, the very low thrust in combination (b) is magnetic field oscillation starts, the oscillating not due to the pressure release. The main reason forces are produced on the fin. seems to be in the two outer magnetic actuators Another interesting chart is the lateral force acting which restrict the ferrofluid to move freely. on the fluid fin. As an example and to have a clear Therefore, the motion of the ferrofluid is limited to view, the lateral force of combination (a) is the space between the two magnetic actuators, as displayed in Fig. 17. observed in Fig. 20.

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As observed in Fig. 21, the 1cm body weighting 0.2kg is displaced more than 0.15 cm with this amount of force. Thus, a structure made of fluid is designed which can be used as a propulsive appendage for the submerged bodies. Encouraged by the presented results, the authors have planed an experimental study on the fluid fins and also, the design of other applications of the fluid structures concept including fluid rudders, fluid propellers, shapeable streamlined high efficiency vessels, and fluid electrical circuits.

5. CONCLUSIONS

In the present study, the feasibility of using a magnetic fluid to form a magnetically controllable and shapeable hydrodynamic propulsive fin for a submerged body is investigated. After reviewing the literature of the mathematical simulation of magnetic fluids and their applications, the concept of “Fluid Structures” and “Fluid Fin” is briefly introduced. Afterward, a validation on the capability of the numerical solver in analyzing the magnetic fluids is conducted against analytical solutions and acceptable errors of 1.21% up to 2.29% are reported. For a better understanding of the performance of the fluid flap or fin in different designs, three different combinations of magnetic actuators are selected. Subsequently, the simulation of the fluid fin is Fig. 20. The restriction of the ferrofluid motion conducted in two stages; the shaping stage and the in combination (b) and the motion direction of actuation stage. In the shaping stage, the shaping the fluid fin. process of the magnetic fluid on the outside surface of the submerged body, using internal magnetic As observed in Fig. 20, when the ferrofluid is actuators, is simulated and the initial stable shape of expected to move upward and downward, the the fluid fin is presented. It is shown that the shape restriction causes it to have a lateral periodic motion of the fluid fin is almost identical to the shape of inward and outward, which highly diminishes the magnetic field isosurface, which implies that produced thrust force. through designing the magnetic field, the shape of the fluid fin can be exactly controlled. In the Regarding the results of the combination (a) and actuation stage, concerning the periodic oscillation assuming that the submerged body weighs 0.2 kg, of the fin, the magnetic actuators are forced to by integrating the force diagram (Fig. 16), the produce an oscillating magnetic field of a 40Hz velocity and displacement of the body during this frequency. It is demonstrated that the oscillating time could be roughly predicted. The velocity and magnetic field makes the ferrofluid oscillate with displacement of the body during from the shaping the same frequency, producing pressure and stage to the actuation stage are illustrated in Fig. velocity changes in the surrounding fluid and thus 2.1. the propulsion force. The estimated average force generated by the fluid fin is up to 0.0158 N in one of the combinations. It is also shown that this amount of force is significant relative to the size of the body and the fin. Finally, it is concluded that a structure made of a shapeable fluid can be used as a fluid appendage for a submerged body.

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